This (clumsy) sentence can be changed to Any adult could do it on their own or -- if you want to avoid the use of their (although I don't understand why) -- it is possible to write Any adult could do it without help. If I wanted to infuriate some members of this forum, I would recommend do it themself, but in most quarters this is regarded as substandard, and do it themselves is preferred.

I'm not clear what Ewie is referring to in #5 -- perhaps it's the phrase do it for themselves.

The use of they as a gender-neutral pronoun with words like anyone, person etc. is well-established. It came into use long before feminists started to use he (or even she) as a politically correct equivalent, which has its own problems.
Example:

"One of my parents is in hospital with a broken leg."
"I'm sorry to hear that. Was it your father or your mother who broke his leg?"
​

Themself is not generally accepted. (Here is an earlier thread, and you can find others by searching for "themself.") Yes, you can find it, but you will seldom find it in edited material. (I loathe it so deeply that it makes my skin crawl.)

I should point out, though, that I am among those holdouts who still doesn't use they, them, their, etc., as singular gender-neutral pronouns. So in your sentence, I'd use himself, which has a long history as a gender-neutral pronoun, or if I only needed to do this once and if it could be done without any awkwardness, I'd write "him or herself." If I had to make several different gender-neutral references, I'd sometimes use himself and sometimes herself.

But if you want to use themselves - and it is increasingly accepted by many if not by me and my fellow holdouts - it is themselves, not themself.

But I meant "do it for themselves" not as "do it themselves". Why this sentence is clumsy?
"I can draw a conclusion for myself, and everyone can do it for themself"

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In post #8 I said that Any adult person could do it her/himself was clumsy. If you don't think it clumsy, try saying it aloud. It also looks odd in print.

You have introduced for themselves (originally you wrote themselves). If it is thought that for themselves is awkward, you might write everyone can do it on their own. The phrase for themselves is separate from the use of they as a neutral gender pronoun.

No, it was the combination of any adult [sing.] + themselves [pl.]. I wouldn't recommend it in formal writing because there are still plenty of people [Hullo, Kate] who object to it

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[Hello, Ewie!]

More importantly, I think there are plenty of English instructors - such as those who grade exams and so on - who object to it. I'm basing this on the number of questions we get on this topic here on the WR forum.

I should point out, though, that I am among those holdouts who still doesn't use they, them, their, etc., as singular gender-neutral pronouns. So in your sentence, I'd use himself, which has a long history as a gender-neutral pronoun, or if I only needed to do this once and if it could be done without any awkwardness, I'd write "him or herself." If I had to make several different gender-neutral references, I'd sometimes use himself and sometimes herself.

But if you want to use themselves - and it increasingly accepted by many if not by me and my fellow holdouts - it is themselves, not themself.

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from a fellow holdout! Just to let the world know we're still alive and kicking!

Keep in mind that I have a strong aversion to themselves used as a singular pronoun, so what I'd write is either "People themselves know what to do" or, if it has to be singular (and sometimes it does), I'd write "The person himself knows what to do."

Those who don't mind using themselves as a singular pronoun would (I think?) use it instead of himself: "The person themselves knows what to do."

But as we've tried to tell you, this is something about which native speakers don't agree. If you don't know how the person who will read this feels about themselves or himself as singular gender-neutral pronouns, I would recommend making it plural.

I'd be happy with either of Kate's two solutions; but in your particular sentence, yakor, I'd go for a third option: simply don't use any reflexive pronoun.

Reflexives for emphasis are generally used to contrast a statement with other implicit (often more likely) possibilities:
- "The President himself came to the celebration" (rather than sending a flunky)
- "I made the cake myself" (rather than getting someone else to do it, or buying it from a shop)

But I can't imagine a situation where anyone else could know something on another person's behalf, so I'd say that "The person knows what to do" says it fully, without the need for an emphatic reflexive.